What is the constant term in the binomial expansion of $(1+3 x)^n\left(1+\frac{1}{3 x}\right)^n$ ?
What is the constant term in the binomial expansion of $(1+3 x)^n\left(1+\frac{1}{3 x}\right)^n$ ?
- $\left(\begin{array}{c}2 n \\ n\end{array}\right)$
- $\left(\begin{array}{c}2 n \\ n-1\end{array}\right)$
- $\left(\begin{array}{c}2 n \\ n+1\end{array}\right)$
- No such term exists
Solution
$
\text { } \begin{aligned}
(1+3 x)^n\left(1+\frac{1}{3 x}\right)^n & =(1+3 x)^n\left(\frac{3 x+1}{3 x}\right)^n \\
& =\frac{(1+3 x)^n(1+3 x)^n}{(3 x)^n} \\
& =\frac{(1+3 x)^{2 n}}{(3 x)^n}
\end{aligned}
$
In $(1+3 x)^{2 n}$ general term is
$
T_{r+1}={ }^{2 n} C_r(3 x)^r
$
$\therefore$ Coefficient of $x^n$ in $(1+3 x)^{2 n}$ is ${ }^{2 n} C_n 3^n$ and coefficient of $x^n$ in $(3 x)^n$ is $3^n$
$
=\frac{{ }^{2 n} C_n 3^n}{3^n}={ }^{2 n} C_n
$
Hence, option (1) is correct
Asked in: AP EAMCET 2020 (22 Sep Shift 2)
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